Leqi Wang

Professor Hubert Bray

Mathematics of Universe 190s

July 17th 2017

Abstract

Enigma was a ciphering machine widely used by Nazi Germans back in

WWII ( the second world war). It was a formidable enemy of Allies of World War II. This

paper will first focus on the Wehrmacht Enigma (M3, with three rotors working and one

plugboard) and then delve deeper into the mechanical structure — how it actually works,

and why the codebreakers devoted so much effort to breaking this machine.

I.The birth of Enigma:

figure 1: the Enigma machine.

!1

Enigma was born at the end of the WWI(World War I). It was invented by a

German engineer Arthur Scherbius and was most wildly used by Nazi Germans in

WWII(World War II) as a protective mechanism of military communication. There were

many different versions of Enigma. Still, military Enigma with a plugboard was the most

complex version.

II.Why we are studying Enigma.

Though Enigma wasn’t the first electronic-mechanical rotor cipher machine

in the history and belonged to the defeated nation Nazi German, and it was not as secure

as modern encrypting machines, what really surprises us is that it was the first cipher

machine broadly put into practice. After the silence in cryptography over a hundred years,

the emergence of Enigma prompted the development in encryption machine. It was

equipped across German armed forces and was engaged in an actual war; it was the

machine that was thoroughly made public and fully explained after the war. It was much

accounted of heads of the state. Hitler himself spoke highly of the security of Enigma, he

claimed Enigma to be “unbreakable” after he was told about the internal settings; Winston

Churchill put breaking Enigma as an “absolute priority”; Dwight Eisenhower gave thank

to Polish mathematicians who made contributions on breaking Enigma in person … …

Here is a popular misunderstanding of Enigma’s fame or notoriety. Though

Enigma itself has nothing to do with the crime Nazi German committed, Enigma is often

related to the rise and fall of the whole empire — the unbreakable character of Enigma

first made it widely used in every military and government services. Thanks to its brilliant

work, the cryptanalysis of the Enigma enabled the western Allies in World War II to read

!2

substantial amounts of secret Morse-coded radio communications of the Axis Powers that

had been enciphered using Enigma machines.

III. The Unbreakable machine.

A M3 Enigma machine has a keyboard, a lamp board, one entry wheel, five

rotors (a scrambler) , one reflector, and a plugboard.

1.input&output

The operator of Enigma will presses the key on the keyboard to implement

input. The corresponding encrypted letter will be shown on the lamp board.

!3

figure2: basic component of Enigma machine

figure3: plugboard of Enigma.

figure4: keyboard figure 5: lamp board

2.Encryption.

-Rotors

In every M3 machine, there are 5 rotors in total but only 3 of which put into

usage. Every time the operator types a letter into Enigma, the letter will first enter

the rotors on the right. And every time the operator taps, the 3rd rotor rotates 1/26 to the

next letter and changes the settings of Enigma:

!

On the opposite side of every rotor, correspondingly, it has 26 electrical contact on the left

side, 26 pin contact on the right side.

rotor3 < − − rotor 2 < − − rotor1

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figure6: a scrambler comprising of 3 rotors figure7: inside the rotor

figure8: electrical contact(left)

The pin and electrical contacts are one-on-one linked by electrical circuit to

make things go faster and more complex(shown in figure 10). For example, if we input

letter A to the scrambler, the current will first get to the pin that represents A. Let’s

assume that this pin is connected with the electrical area that represents P. When the letter

comes out from the first rotor, A has already become P. So all the letter has to do is to

repeat this process for 3 times.

And how this will work out is indicated in the window:

!5

figure9: pin contact (right) figure10: one-on-one connection between electrical and pin contact.

figure11: rotor windows of M4 machine of navy

Still, the settings eventually depend on the internal electrical circuit of Enigma.

-reflector

The reflector is sitting behind the last rotor. It has 26 pin contact areas

connected with their electrical counterpart on the third rotor. And inside every reflector,

the pins are connected by pairs. If one letter enters the reflector, it’ll come out from the

other side as another letter. And the letter it becomes will go back into the rotors and

come out from the first rotor, enter the plugboard, and eventually shown on the lamp

board.

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figure13:How Enigma works in general

figure12: reflector

Generally speaking, the reflector is a mirror, but it does not reflect the same things as

input. For example, If I input the first letter into Enigma.

and here is the problem, when D enters the reflector, it’ll become something else.

Therefore, we can safely say that whatever goes into Enigma machine,

when it comes out, it cannot be itself. This feature becomes the most important flaw of

Enigma. And the codebreakers of the Allies finally beat Enigma by this flaw. And I will

go back into this problem in my next paper.

-plugboard(figure3)

The essence of plugboard is working on the principle of Mono-alphabetic

substitution, which means the plaintext and ciphertext have the same amount of letters

and these letters are also corresponding. This relationship is shown in figure13. If letter A

is encrypted into E, then the plaintext of E must be A.

So, how many combinations can be brought by plugboard?

First, two letters are connected by one wire, which means every wire concatenates//

connects 2 letters. Once one wire is plugged in, the available letters left will reduce by 2.

The total number left is

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A—>first rotor—>B B—>second rotor—>C C—>third rotor —>D

D—>reflector—>E

!

And the total possibility of plugboard is actually the product of these processes.

We choose 2n letters from 26 as the number of letters we want to change,

we have:

!

kinds of situations.

Second,

(2n-1)*(2n-3)*… …*1

is the number of pairs these 2n letters can become.

Therefore, the total number of variation is the product of these two formula,

which turns out to be:

!

So this is the possibility of plugging on n wires on the plugboard:

2n − 2

26!(26 − 2n)! ⋅ (2n)!

26!(26 − 2n)! ⋅ 2n ⋅ n!

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n = 0, Possibility = 1 n = 1, P = 325 n = 2, P = 44,850 n = 3, P = 3,453,450 n = 4, P = 164,038,875 n = 5, P = 5,019,589,575 n = 6, P = 100,391,791,500 n = 7, P = 1,305,093,289,500 n = 8, P = 10,767.019,638,375 n = 9, P = 58,835.098,191,875 n = 10, P = 150,738,274,937,250

The possibility will come to maximum when there are 11 wires plugged in, meaning 22

letters paired up. But in the real M3 settings, German made only 20 jacks and that’s

150,738,274,937,250 settings at most.

-The number of combinations in Enigma

Since Every rotor has 26 independent circuit connection from each pin to

each electrical contact. And each connection corresponds with one conversion of a letter,

which means, one rotor will have 26 starting points, 26 different possibilities.（ Since the

second rotor is influenced by the marching problem, its period only contains 25

possibilities, but we’ll dismiss this difference.） Every time the typer input one letter, the

rotor on the right side will rotate to next letter(1/26). So there are

!

total possible starting point configurations.

And since there are 5 rotors in total and only three at word, we have to

randomly pick up 3 rotors from 5 possible rotors. So there are

!

26*26*26 = 17576

5!(5 − 3)!

= 60

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n = 11, P = 205,552,193,096,250（MAX) n = 12, P = 102,776,096,548,125 n = 13,P = 7,905,853,580,625

possible combinations(The possibilities in Enigma is actually less than 17567 but only

16900, since the second rotor only has 25 starting points other than 26, so the formula

should be 26*25*26).

As a result, there are

!

possible settings in a M3 scrambler.

In 1930, the German army introduced the plugboard that further secure the

machine from brute-force attack to Enigma machine. And inside the plugboard, there are

10 pairs of alphabets being connected. Two letters in one pair can be swapped over. This

level of scrambling only appears in military Enigma machine and adds the most

combinations in to Enigma settings. So there are 26 letters in alphabets, and we have

!

different ways to arrange these letters. However, we don’t have to get every combination

of these 26 letters because in Enigma there are only 10 pairs of letters, 20 letters in total,

being swapped over. Therefore we have to divide 26! by 6! so that we can dismiss the

combination of those 6 letters we don’t care about. And the formula becomes

!

possibilities. After that, since we don’t know about the arrangement of these 10 pairs of

letters, we don’t care about them. Therefore, we can divided by 10!:

!

There is the last factor we have to divide by: since there are two letters in a pair, the order

of these letters does not matter because they will be swapped over anyhow(B and A is the

17576*60 = 1054560

26!

26!/6!

26!/6!/10!

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same as A and B). So each pair can divided by 2, and there are 10 pairs , so we divide by

2^10: 1

!

So there are 150738274937250 possible ways of setting up in a plugboard.

And in total, there are

kinds of Enigma M3 combinations, approximately 159 with 18 zeros behind.

If we had ten people checking one setting a minute for 24 hours a day and 7 days a

week, it’ll take

!

to work out all settings in Enigma.

-Turning Point

Though Enigma machine seemed unbreakable, the brilliant mind in the Allied country

solved this hard puzzle by combining their effort: the Polish mathematicians used

permutation matrix and a machine called “Bomba” worked out the settings of 3-rotor

Enigma machine. And after the German-Soviet invasion to Poland, British

26!(26 − 20)! ⋅ 210 ⋅ 10!

= 150,738, 274,937, 250

5!(5 − 3)!

⋅ 263 ⋅26!

(26 − 20)! ⋅ 210 ⋅ 10!= 158,962, 555,217, 826,360, 000

20MillionYears

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mathematicians, Dilly Knox, Alan Turing and Gordon Welchman, gradually made effort

to completely decipher all versions of Enigma machines.

Throughout the second world war, though there were only three kinds of ciphering

machines: Enigma, Lorenz SZ40/SZ42 and SIEMENS AG T43/T52. And Enigma was the

most widely used across the both on map and the timeline — nineteen years. But this

immoderate use eventually made Nazi Germans lose the entire war, and this was, and still

is, the greatest flaw in the Unitary Security System. “The first weakness, the machine

design itself, was a minor one. When Enigma enciphered a letter, it was guaranteed that

the resulting enciphered letter would not be the same letter. For example, ‘A’ could be

translated into any letter other than ‘A’, ‘B’ would never be enciphered as ‘B’, etc. By

itself would not have allowed the Allies to read Enigma ciphertext but it did assist

cryptographers in finding the human errors. The more important weakness was the

human one. The mistakes made by the Enigma operators and the poor procedures put

into place by the German military were legion; without them the Poles and the British

would have had no hope of cracking them. Gordon Welchman, one of the officers in the

British program, said ‘the machine as it was would have been impregnable if it had been

used properly’ but pointed out twelve serious errors in procedure that, if corrected, would

‘have stopped us cold.’”

Conclusion

In the world of cryptography, Enigma is absolutely a huge breakthrough. Its

contribution should be counted as two mileposts: one for inventing this machine, which is

a huge intelligence challenge put on Allies of WWII and on the whole human race.

“Breaking Enigma, the historian estimated, shortened the war by more than two years,

saving over 14 million lives.”

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Another mileposts is for the difference Enigma made in science by studying and solving

it, which generated an entire new scientific field — computer science.

————————————————————————————————

————

Reference

Yanfeng Zhao. (2008).Legends of Enigma. Beijing, China: Science publishing house.

Andrew Hodge. Counting the Possible Plugboard Settings.https://

www.codesandciphers.org.uk/enigma/steckercount.htm(Accessed July17th, 2017)

Cryptanalysis of the Enigma.(2017, July 13). In Wikipedia, the free encyclopedia.

Retrieved July 17th, 2017, from https://en.wikipedia.org/wiki/

Cryptanalysis_of_the_Enigma

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17th, 2017, from https://zh.wikipedia.org/wiki/

%E6%81%A9%E5%B0%BC%E6%A0%BC%E7%8E%9B%E5%AF%86%E7%A0%81

%E6%9C%BA#.E5.8F.8D.E5.B0.84.E5.99.A8

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www.zhihu.com/question/28397034 (Accessed July 17th, 2017)

!13

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(Accessed July 16th, 2017)

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tide-of-wwii-11363990654704 (Accessed July 16th, 2017)

Numberphile.(2013, Jan 10). 158,962,555,217,826,360,000 (Enigma Machine) -

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2017)

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